Probability and Mathematical Physics Seminar

Synopsis:

Large sample covariance matrices (SCM) are not the best estimates of true covariances. Using tools from random matrix theory (RMT) and free probability one can compute the eigenvalue spectrum of the SCM from independent or even auto-correlated samples. One can also compute the optimal estimator of true covariance from sample data under a natural hypothesis of absence of prior knowledge about eigenvectors. This estimator can be expressed in RMT language but is easier to understand in the optimization/validation (O/V) framework used in machine learning. A recent leave-one-out algorithm is actually the best numerical implementation of the RMT optimal estimator.

Functions of the true covariance matrix (such as its inverse) can also be estimated using the same schemes. I speculate that the link between the RMT and O/V frameworks can help us distinguish between signal and noise in very complex noisy data sets such as neural recording data. References: Bun, Bouchaud and Potters, Physics Reports 666 (2017), forthcoming book by Potters and Bouchaud, Cambridge (2020).

Speaker Bio: Marc Potters is Chief Investment Officer of CFM, an investment firm based in Paris. Together with Jean-Philippe Bouchaud, he supervises the research team with particular focus on developing concrete applications in financial forecasting, portfolio construction, risk control and execution. Marc maintains strong links with academia and as an expert in Random Matrix Theory has taught at UCLA and Sorbonne University. Marc obtained his PhD in physics from Princeton University and joined CFM in October 1995 as a researcher in quantitative finance. Marc continues to publish papers in statistical finance with his research team and co-authored ‘Theory of Financial Risk and Derivative Pricing’ with Jean-Philippe.

Notes:

Cross-posting from the MaD seminar.

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Columbia/Courant joint seminar

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Understanding the distribution of the spectrum of a random matrix as the dimension grows is one of the main problems in random matrix theory. This includes, among others, the study of the limiting spectral distribution and the behavior at the boundary of the support of the limiting measure. It is known that the empirical spectral distribution of a square random matrix (resp. symmetric) with i.i.d centered entries with unit variance converges to the circular law (resp. semi-circular) as the dimension grows. 
In this talk, we are interested in the stability of these results and the behavior of the spectrum when the i.i.d assumption is relaxed. Random graphs provide models encapsulating sparsity and dependence. 
The talk will investigate:
1-The limiting spectral distribution of random regular graphs,
2-The behavior of the extreme eigenvalues/singular values and the spectral gap of random graphs. 

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We consider a continuous time random walk on the rooted binary tree of depth \(n\) with all transition rates equal to one and study its cover time, namely the time until all vertices of the tree have been visited. We prove that, normalized by \(2^{n+1} n\) and then centered by \((\log 2) n - \log n\), the cover time admits a weak limit as the depth of the tree tends to infinity. The limiting distribution is identified as that of a Gumbel random variable with rate one, shifted randomly by the logarithm of the sum of the limits of the derivative martingales associated with two negatively correlated discrete Gaussian free fields on the infinite version of the tree. The existence of the limit and its overall form were conjectured in the literature. Our approach is quite different from those taken in earlier works on this subject and relies in great part on a comparison with the extremal landscape of the discrete Gaussian free field on the tree. Joint work with Aser Cortines and Santiago Saglietti.

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A key question in population biology is understanding the conditions under which the species of an ecosystem persist or go extinct. Theoretical and empirical studies have shown that persistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of n interacting species that live in a stochastic environment. Our models are described by n dimensional piecewise deterministic Markov processes. These are processes (X(t), r(t)) where the vector X denotes the density of the n species and r(t) is a finite state space process which keeps track of the environment. In any fixed environment the process follows the flow given by a system of ordinary differential equations. The randomness comes from the changes or switches in the environment, which happen at random times. We give sharp conditions under which the populations persist as well as conditions under which some populations go extinct exponentially fast. As an example we show how the random switching can `rescue' species from extinction. The talk is based on joint work with Dang H. Nguyen (University of Alabama).

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We discuss a derivation of a continuum mean-curvature flow as a scaling limit of a class of particle systems, more robust than previous methods. We consider zero-range + Glauber interacting particle systems, where the zero-range part moves particles while preserving particle numbers, and the Glauber part allows creation and annihilation of particles. When the two parts are simultaneously seen in certain (different) time-scales, and the Glauber part is "bi-stable", a mean-curvature flow can be captured directly as a limit of the mass empirical density.

Such a "direct" limit might be compared with a "two-stage" approach: When the zero-range part is diffusively scaled but the Glauber part is not scaled, the hydrodynamic limit is a non-linear Allen-Cahn reaction-diffusion PDE. It is well-known in such PDEs, when the "bi-stable" reaction term is now scaled, that the limit of the solutions takes on stable values across an interface moving by a mean-curvature flow. This is joint work-in-progress with Tadahisa Funaki and Danielle Hilhorst.

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Right and left eigenvectors of non-Hermitian matrices form a bi-orthogonal system, to which one can associate homogeneous quantities known as overlaps. The matrix of overlaps is Hermitian and positive-definite; it quantifies the stability of the spectrum, and characterizes the joint eigenvalues increments under Dyson-type dynamics. These variables first appeared in the physics literature, when Chalker and Mehlig calculated their conditional expectation for complex Ginibre matrices (1998). For the same model, we extend their results by deriving the distribution of the overlaps and their correlations (joint work with P. Bourgade). Similar results are expected to hold in other integrable models, and some have been established for quaternionic Gaussian matrices.

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I will explain some current work on a stochastic interacting particle system, branching Brownian motion with selection, and its hydrodynamic limit, which is a free boundary PDE problem. At each branch event in the branching Brownian motion, a particle is removed from the system according to a fitness function, so that the total number of particles, N, is preserved. It is interesting to understand how this selection process effects the evolution of the ensemble of particles. De Masi, Ferrari, Presutti, Soprano-Loto recently showed that in one space dimension, when the left-most particle is always selected, then as N grows the particle system converges to the solution of a certain parabolic free boundary problem which has traveling wave solutions -- this scenario corresponds to a fitness function which is monotone. In joint work with Julien Berestycki, Éric Brunet, Sarah Penington, we study this problem in higher dimensions with a fitness function that has compact level sets. The hydrodynamic limit (large N limit) is also a parabolic free boundary problem, related to the parabolic obstacle problem. The solution of this PDE problem converges, in the large time limit, to an eigenfunction of the laplacian. With Erin Beckman, we also study the problem in 1-d with non-monotone fitness, which leads to a kind of pulsating traveling wave behavior and a metastability phenomenon depending on the fitness function.

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Free probability provides a unifying framework for studying random multi-matrix models in the large \(N\) limit. Typically, the purview of these techniques is limited to invariant or mean-field ensembles. Nevertheless, we show that random band matrices fit quite naturally in this framework. Our considerations extend to the infinitesimal level, where finer results can be stated for the \(\frac{1}{N}\) correction. As an application, we consider the question of outliers for localized perturbations of our model.

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I will develop a framework which is particularly pathwise in nature to show a Random Dynamics systems has at most one long time statistical behavior. Its pathwise nature is particularly well adapted to singular (roughly forced) SPDEs. 

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The Ricci flow on a surface is an intrinsic evolution of the metric converging to a constant curvature metric within the conformal class. It can be seen as an (infinite-dimensional) gradient flow. We introduce a natural 'Langevinization' of that flow, thus constructing an SPDE with invariant measure expressed in terms of Liouville Conformal Field Theory.

Joint work with Hao Shen (Wisconsin).

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We will present new coupling techniques for analyzing ergodicity of nonlinear stochastic PDEs with additive forcing. These methods complement the Hairer-Mattingly approach (2006, 2011). In the first part of the talk, we demonstrate how a generalized coupling approach can be used to study ergodicity for a broad class of nonlinear SPDEs, including 2D stochastic Navier-Stokes equations. This extends the results of [N. Glatt-Holtz, J. Mattingly, G. Richards, 2017]. The second part of the talk is devoted to SPDEs that satisfy comparison principle (e.g., stochastic reaction-diffusion equation). Using a new version of the coupling method, we establish exponential ergodicity of such SPDEs in the hypoelliptic setting and show how the corresponding Hairer-Mattingly results can be refined.

O. Butkovsky, A. Kulik, M. Scheutzow (2018). Generalized couplings and ergodic rates for SPDEs and other Markov models. arXiv:1806.00395; to appear in "The Annals of Applied Probability".

(Joint work with Alexey Kulik and Michael Scheutzow)

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The hierarchical Laplacian was initially introduced in the works of N. Bogolubov and his school (V. Vladimirov, I. Volovich, E. Zelinov) as an essential object in the \(p\)–adic analysis. Similar ideas were developed by F. Dyson in his famous paper on the phase transitions in \(1D\) Ising model with the long range potentials.

We define Dyson–Vladimirov hierarchical Laplacian \(\Delta\) as the non-local operator in \(L^2 (\mathbb{R}, dx)\) associated the Dyson metric on \(\mathbb{R}\). Such Laplacian has many features of the classical fractals (renorm group etc.).

The talk will present the elements of the spectral theory of the hierarchical Hamiltonian \(H = -\Delta + V(x)\). The theory includes the standard results (on the essential self-adjointness, negative spectrum etc.) for the deterministic operators and the results in the spirit of the Anderson localization for the class of the random Schrödinger operators.

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Neural networks have experienced a renaissance in recent years, finding success in tasks from machine vision (e.g. self-driving cars) to natural language processing (e.g. Alexa or Siri) and reinforcement learning (e.g. AlphaGo). A mathematical theory of how and why they work is only in the very starting stages.

The purpose of this talk is to address an important numerical stability issue for neural networks, known as the exploding and vanishing gradient problem. I will explain what this problem is and how it turns precisely into a problem of studying products of many matrices in the regime where both the sizes and the number of matrices tend to infinity together. I will present some joint work with Mihai Nica on the behavior of matrices in this regime.

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Critical percolation is fairly well-understood on \(Z^d\) for \(d > 11\). Exact values of many critical exponents are rigorously known: for instance, the “one-arm” probability that the origin is connected by an open path to distance \(r\) scales as \(r^{-2}\). However, most existing methods rely heavily on the symmetries of the lattice, so they do not extend to fractional spaces. We will discuss progress on these questions in the high-dimensional upper half-space (and within cubes), including the result that the half-space one-arm probability scales as \(r^{-3}\).

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